A graph G is arbitrarily partitionable (AP for short) if for every partition (tau_1, ..., tau_p) of |V(G)| there exists a partition (V_1, ..., V_p) of V(G) such that each V_i induces a connected subgraph of G with order tau_i. If, additionally, each of k of these subgraphs contains an arbitrary vertex of G prescribed beforehand, then G is arbitrarily partitionable under k prescriptions (AP+k for short). It is known that AP+k-graphs on n vertices are (k+1)-connected, and have thus at least n(k+1)/2 edges. We show that there exist AP+k-graphs on n vertices and n(k+1)/2 edges for every k >= 1 and n >= k.